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What do you think of when you hear the words Sinn Féin? Outside Ireland, most would give the same two-word answer: Gerry Adams.

The godfather of modern Irish republicanism has retired from front-line politics, but any mention of the party he led for 35 years evokes a familiar image – bushy beard, thick glasses, and an even thicker Belfast accent that was banned from broadcast and clumsily overdubbed on British television for so long.

Nobody is more conscious of this than Mary Lou McDonald, the Dubliner who succeeded Adams unopposed as leader of the IRA’s former political wing in February.

Hers is an unenviable political inheritance. “Gerry Adams is the most recognisable Irishman on the planet,” McDonald admitted when we met recently in her office in Leinster House, the seat of Ireland’s parliament. “Sorry Bono, but that’s the way it is, right?”

McDonald, universally known as “Mary Lou”, is nonetheless busy reshaping Sinn Féin in her own image. She appears to be succeeding: in the Republic, the party has enjoyed a rise in the polls since her long-expected ascension.

The 49-year-old’s biography is nothing like that of her northern predecessor. She went to a private school in an affluent south Dublin suburb and then to Trinity College. She was never involved in IRA terrorism, and, of course, is a woman, as is her deputy Michelle O’Neill.

“The truth is, my friends, I won’t fill Gerry’s shoes,” McDonald told party members upon her appointment. “But the news is that I brought my own.” Recalling the line when we meet, she gestures to her high-heeled boots: “Exhibit A.” Commentators and rivals say she is in thrall to the party’s old guard; that Adams and IRA alumni still pull the strings. “I don’t have to say, or even explain how misogynistic that is,” McDonald sighs.

Where Adams had something of an eldritch quality about him, McDonald is warm. When I tell her my grandparents are from Dublin, she offers me a high-five. Even allowing for his eccentric social media presence, it’s hard to imagine her predecessor offering a journalist with an English accent the same treatment.

The style has certainly changed, but what of the substance? Despite a series of tight Brexit votes in the House of Commons the night before we meet, the perennial question of whether Sinn Féin will take its seven seats elicits that familiar answer: a flat, but patient, no. “There isn’t an anti-Brexit make-it-stop critical mass at Westminster,” she says – adding that she is no fan of Theresa May, but Jeremy Corbyn as prime minister would be “a great thing”.

Plus ça change. With Irish issues at the heart of the argument on Brexit, Sinn Féin’s solution – a united Ireland – is the same as it has been for decades, and has increasing currency beyond the republican movement.

But McDonald also wants to lead the party somewhere it has never been before: into government in Dublin.

Once an also-ran, Sinn Féin is now the third-largest party in the Dáil Éireann and will hold the balance of power at the next election. Neither Leo Varadkar, the Taoiseach and leader of Fine Gael, or Micheál Martin, leader of Fianna Fáil, say they will speak to her – but McDonald rejects the threats of ostracism: “Nobody’s going to tell the people who vote for us that somehow the people that they elect are to be excluded forever from serving in government,” she says.

Her everywoman appeal to middle Ireland is something the party has hitherto lacked. McDonald was at the forefront of the successful campaign to remove the constitutional bar on abortion, yet is conscious that “sexy” equality issues can overshadow less fashionable causes.

Making the leap from left-populist third force to mainstream party of government will not be easy, however. Agreeing a programme for government with Varadkar, whom McDonald has criticised as “smarmy”, would be like trying to broker a coalition deal between Margaret Thatcher and Tony Benn. And although Brexit has made the question of a united Ireland “more immediate”, knotty challenges lie ahead.

Sharing the stage with other parties in a unity referendum could be particularly problematic: “We will have to give the issues away to make it happen.” Compared to the NHS, the Irish health system is “banjaxed”, she observes, and understands why some Northern nationalists would sooner stay in the UK than “pay 60 quid to see their GP”.

Can Sinn Féin’s uneasy coalition between rural Catholicism in the north and metropolitan leftism in the south hold? She thinks so. “I’m not from outer Kathmandu. I’m from Dublin. Do you know what I mean?”

Then there is the question of how to make Unionists such as Arlene Foster feel at home in her “new Ireland”. Would she be open to changing the flag, the anthem? “Everything needs to be on the table.”

For either of her two missions to succeed, Sinn Féin and its new generation of leaders must define the future. But the party’s past casts a long shadow, and the scars caused by violent, “physical force” republicanism and the Troubles run deep.

“Life is about change: things move on, politics move on, families move on,” McDonald says (though IRA victims, some of whom she has met, fiercely dispute that). Then she adds: “We shouldn’t be asking people [on any side] to disavow their tradition.” And so indeed she isn’t. Instead, she is creating her own. Call it emotional-force republicanism.